Continuity of certain differentials on finitely augmented Teichmüller spaces and variational formulas of Schiffer-Spencer's type
type: Item
instance of: scholarly article
title: Continuity of certain differentials on finitely augmented Teichmüller spaces and variational formulas of Schiffer-Spencer's type
zbMATH Open document ID: 0606.32013
author: Masahiko Taniguchi
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1986-01-01T00:00:00Z
review text: The author extends some of his earlier work [see J. Math. Kyoto Univ. 25, 597-617 (1985; Zbl 0597.30056)] on the continuity properties of holomorphic and harmonic differentials on Riemann surfaces. The Riemann surfaces are permitted to have nodes hence the surfaces R vary over the augmented Teichmüller spaces [see the reviewer, Ann. Math., II. Ser. 105, 29-44 (1977; Zbl 0347.32010) for the relevant topologies]. Given a one parameter family \(\{R_ t| t>0\}\) of nonsingular Riemann surfaces of finite conformal type, there is often a marking preserving deformation \(f_ t: R_ t\to R_ 0\) which collapses curves into nodes. The family \(R_ t\) may be chosen so that each \(f_ t\) is conformal away from a neighborhood of the nodes. Under these conditions, the author finds that, when normalizations are respected, the Green's function \(g_ t\) converge uniformly to a Green's function \(g_ 0\) as \(t\to 0\). Here we assume that \(g_ 0\) exists. If \[ \phi (q;R)=idq(\cdot,q;R) - *dg(\cdot,q;R) \] then \(\phi (f_ t^{-1}(q);R)\to \phi (q;R)\) in the Dirichlet norm \(\iint \omega \wedge *\omega.\) Variational formulas of Schiffer-Spencer type are also presented.
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 3979506
zbMATH Keywords: continuity properties of holomorphic and harmonic differentials on Riemann surfaces
zbMATH Keywords: augmented Teichmüller spaces
zbMATH Keywords: marking preserving deformation
zbMATH Keywords: Variational formulas of Schiffer-Spencer type
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228494
OpenAlex ID: W2046283134
cites work: Q5719551
cites work: Q5685358
cites work: Q5832232
cites work: Remarks on topologies associated with squeezing a non-dividing loop on compact Riemann surfaces
cites work: Dirichlet finite harmonic differentials with integral periods on arbitrary Riemann surfaces
cites work: On convergence of holomorphic abelian differentials on the Teichmüller spaces of arbitrary Riemann surfaces
cites work: Square integrable harmonic differentials on arbitrary Riemann surfaces with a finite number of nodes
cites work: Q3749304
cites work: Variational formulas on arbitrary Riemann surfaces under pinching deformation
DOI: 10.2748/TMJ/1178228494
description: scientific article; zbMATH DE number 3979506